Definitions of integral over a bounded set.

[gotjrgkr]

Hi!
I want to learn a course of "general relativity".
For this, I've realized that I have to master the differential geometry.
So, I've chosen Lee's book called " introduction to smooth manifolds".
In the appendix of the book, some required knowledege of integrations on an euclidean space are listed. And I found a definition of integral over a bounded set.
Suddenly, a question has passed through my brain. I've also tried to learn an integration theory from fleming's book " functions of several variables". But, it seems that the book uses a different definition from lee's book. For example, fleming's integrates a function over a bounded measurable set. On the other hand, lee's book doesn't care about measurable set. the lee's book integrates a function over a just bouded set.
This is the point where my problems occur. I think the definitions of integral over a bouded set must be equivalent. But as you can see, they are different.
I can't understand why authors don't use a common definition of integration over a bouded set. Do I have to follow the each different definition of a integration of different books??
Then, I believe that it would be difficult to exchange common ideas when talking about something of math.
Why do some authurs use one definition of integration and others use others??
Do you think that all following books; munkres's "Analysis on manifolds", spivak's "calculus on manifolds", fleming's "functions of several variables", and so on, use the equivalent definitions of integration over a bounded set??
How do you think about my problems?
What is the wrong parts that I've mentioned above??
Please, notice about them for me...
This is very serious problem for me.

 

[HallsofIvy]

I suspect that Lee is using the "Riemann integral" rather than the "Lesbeque integral" of Fleming. Using a Riemann integral sharply limits the sets over which a function can be integrated- a much stronger condition than merely "measurable". For example, the set of all irrational numbers between 0 and 1 is measurable (and has measure 1) so that we can integrate a Lesbeque integral over it but it contains no interval so we would not be able to integrate a Riemann integral over it.

 

[gotjrgkr]

I want to learn the whole story of integral.
I've heard that "Riemann integral" was developed earier than "Lebesgue integral".
I'd like to ask you following questions...

1.
You said [Using a Riemann integral sharply limits the sets over which a function can be integrated- a much stronger condition than merely "measurable". ].
From this, can I accept it to imply that "Riemann integral" can be defined by "Lebesgue integral", so that "Riemann integral" is just a kind of "Lebesgue integral"??

2.
Do you mean that the "Riemann integral" integrates a function over a multi-dimensional rectangle??
3.
In the lee's appendix, munkres's analysis on manifolds,etc, they introduce a definition of integrating a function over a bounded set by using "Riemann integral" over a rectangle.
On the other hand, in fleming's book and other materials dealing with the lebesgue theory, they define an integration of a function over a bounded "measurable" set.
As you can see, I think, the former one is dealing wider class than the latter in that it integrates any bounded set, but the latter integrates only a bounded "measurable" set.
I don't get an idea why there're two defintions of integrating a bounded set. How are they related with each other??.why does it seem to me that the former one is wider than the former? isn't it strange if considering that the former one is defined by "Riemann integral" and the latter is defined by "Lebesgue integral"?